Modified Helmholtz closed spectral one-form ($W=e^{-ikz-\lambda \bar{z}/(ik)}[(q_z+ikq)dz-(q_{\bar{z}}+\lambda q/(ik))d\bar{z}]$)

For $k\ne 0$, differentiation gives $dW=-2e^{-ikz-\lambda \bar{z}/(ik)}(q_{z\bar{z}}-\lambda q)dz\wedge d\bar{z}$. Thus the differential form is closed precisely for the modified Helmholtz equation $\Delta q-4\lambda q=0$. Integrating around an oriented domain gives a global relation. The primitive $\mu e^{-ikz-\lambda \bar{z}/(ik)}$ also produces compatible first-order equations whose spectral jumps reconstruct $q$.